1. Plane EM Wave: Oblique Incidence
𝐸 ⊥ 𝑟 = 𝐸 0⊥ 𝑟 𝑒 −𝑗 𝛽 𝑟 ∙ 𝑟
𝐸 ⊥ 𝑖 = 𝐸 0⊥ 𝑖 𝑒 −𝑗 𝛽 𝑖 ∙ 𝑟
𝐸 || 𝑖 = 𝐸 0|| 𝑖 𝑒 −𝑗 𝛽 𝑖 ∙ 𝑟
𝐸 || 𝑟 = 𝐸 0|| 𝑟 𝑒 −𝑗 𝛽 𝑟 ∙ 𝑟
Plane of Incidence
𝐸 ⊥ 𝑡 = 𝐸 0⊥ 𝑡 𝑒 −𝑗 𝛽 𝑡 ∙ 𝑟
0°≤ 𝜃 𝑖 , 𝜃 𝑟 , 𝜃 𝑡 ≤90°
𝐸 || 𝑡 = 𝐸 0|| 𝑡 𝑒 −𝑗 𝛽 𝑡 ∙ 𝑟

2. Propagation Directions and Magnitude Relations
𝐸 0⊥ 𝑟 𝐸 0⊥ 𝑖 = 𝜂 𝑡 cos 𝜃 𝑖 − 𝜂 𝑖 cos 𝜃 𝑡 𝜂 𝑡 cos 𝜃 𝑖 + 𝜂 𝑖 cos 𝜃 𝑡
𝐸 0⊥ 𝑡 𝐸 0⊥ 𝑖 = 2 𝜂 𝑡 cos 𝜃 𝑖 𝜂 𝑡 cos 𝜃 𝑖 + 𝜂 𝑖 cos 𝜃 𝑡
𝐸 0|| 𝑟 𝐸 0|| 𝑖 = 𝜂 𝑖 cos 𝜃 𝑖 − 𝜂 𝑡 cos 𝜃 𝑡 𝜂 𝑖 cos 𝜃 𝑖 + 𝜂 𝑡 cos 𝜃 𝑡
𝐸 0|| 𝑡 𝐸 0|| 𝑖 = 2 𝜂 𝑡 cos 𝜃 𝑖 𝜂 𝑖 cos 𝜃 𝑖 + 𝜂 𝑡 cos 𝜃 𝑡
𝜃 𝑖 = 𝜃 𝑟
𝑛= 𝜀 𝑟
𝑛: index of refraction
𝜀 𝑖 𝜇 𝑖 sin 𝜃 𝑖 = 𝜀 𝑡 𝜇 𝑡 sin 𝜃 𝑡
𝜀 𝑟 : relative permittivity
𝑛 𝑖 sin 𝜃 𝑖 = 𝑛 𝑡 sin 𝜃 𝑡

3. Reflection and Transmission Coefficients
Γ || =− 𝐸 0|| 𝑟 𝐸 0|| 𝑖
Γ ⊥ = 𝐸 0⊥ 𝑟 𝐸 0⊥ 𝑖
𝑇 || = 𝐸 0|| 𝑡 𝐸 0|| 𝑖
𝑇 ⊥ = 𝐸 0⊥ 𝑡 𝐸 0⊥ 𝑖

4. Reflection and Transmission Coefficients
Γ ⊥ = 𝐸 0⊥ 𝑟 𝐸 0⊥ 𝑖 = 𝜂 𝑡 cos 𝜃 𝑖 − 𝜂 𝑖 cos 𝜃 𝑡 𝜂 𝑡 cos 𝜃 𝑖 + 𝜂 𝑖 cos 𝜃 𝑡
Γ || =− 𝐸 0|| 𝑟 𝐸 0|| 𝑖 =− 𝜂 𝑖 cos 𝜃 𝑖 − 𝜂 𝑡 cos 𝜃 𝑡 𝜂 𝑖 cos 𝜃 𝑖 + 𝜂 𝑡 cos 𝜃 𝑡
𝑇 ⊥ = 𝐸 0⊥ 𝑡 𝐸 0⊥ 𝑖 = 2 𝜂 𝑡 cos 𝜃 𝑖 𝜂 𝑡 cos 𝜃 𝑖 + 𝜂 𝑖 cos 𝜃 𝑡
𝑇 || = 𝐸 0|| 𝑡 𝐸 0|| 𝑖 = 2 𝜂 𝑡 cos 𝜃 𝑖 𝜂 𝑖 cos 𝜃 𝑖 + 𝜂 𝑡 cos 𝜃 𝑡

5. Reflection and Transmission Coefficients
Γ ⊥ = 𝐸 0⊥ 𝑟 𝐸 0⊥ 𝑖 = 𝑛 𝑖 cos 𝜃 𝑖 − 𝑛 𝑡 cos 𝜃 𝑡 𝑛 𝑖 cos 𝜃 𝑖 + 𝑛 𝑡 cos 𝜃 𝑡
Γ || =− 𝐸 0|| 𝑟 𝐸 0|| 𝑖 =− 𝑛 𝑡 cos 𝜃 𝑖 − 𝑛 𝑖 cos 𝜃 𝑡 𝑛 𝑡 cos 𝜃 𝑖 + 𝑛 𝑖 cos 𝜃 𝑡
𝑇 ⊥ = 𝐸 0⊥ 𝑡 𝐸 0⊥ 𝑖 = 2 𝑛 𝑖 cos 𝜃 𝑖 𝑛 𝑖 cos 𝜃 𝑖 + 𝑛 𝑡 cos 𝜃 𝑡
𝑇 || = 𝐸 0|| 𝑡 𝐸 0|| 𝑖 = 2 𝑛 𝑖 cos 𝜃 𝑖 𝑛 𝑡 cos 𝜃 𝑖 + 𝑛 𝑖 cos 𝜃 𝑡

6. Reflection and Transmission Coefficients
Example 1: air to water
𝑛 𝑖 =1, 𝑛 𝑡 =4/3
Brewster angle
𝜃 𝐵

7. Reflection and Transmission Coefficients
Example 2: water to air
𝑛 𝑖 =4/3, 𝑛 𝑡 =1
𝜃 𝑖 > 𝜃 𝑐
𝜃 𝑐
𝜃 𝐵
Brewster angle
Critical angle

8. Total (Internal) Reflection: Critical Angle
Γ ⊥ = 𝐸 0⊥ 𝑟 𝐸 0⊥ 𝑖 = 𝑛 𝑖 cos 𝜃 𝑖 − 𝑛 𝑡 cos 𝜃 𝑡 𝑛 𝑖 cos 𝜃 𝑖 + 𝑛 𝑡 cos 𝜃 𝑡
Γ || =− 𝐸 0|| 𝑟 𝐸 0|| 𝑖 =− 𝑛 𝑡 cos 𝜃 𝑖 − 𝑛 𝑖 cos 𝜃 𝑡 𝑛 𝑡 cos 𝜃 𝑖 + 𝑛 𝑖 cos 𝜃 𝑡
𝜃 𝑖 = 𝜃 𝑐
Critical angle
𝜃 𝑡 =90°
Γ ⊥ =1
Γ || =−1
𝑛 𝑖 sin 𝜃 𝑖 = 𝑛 𝑡 sin 𝜃 𝑡
sin 𝜃 𝑐 = 𝑛 𝑡 / 𝑛 𝑖
𝑛 𝑖 > 𝑛 𝑡

9. Total Transmission: Brewster Angle
Γ ⊥ = 𝐸 0⊥ 𝑟 𝐸 0⊥ 𝑖 = 𝑛 𝑖 cos 𝜃 𝑖 − 𝑛 𝑡 cos 𝜃 𝑡 𝑛 𝑖 cos 𝜃 𝑖 + 𝑛 𝑡 cos 𝜃 𝑡
Γ || =− 𝐸 0|| 𝑟 𝐸 0|| 𝑖 =− 𝑛 𝑡 cos 𝜃 𝑖 − 𝑛 𝑖 cos 𝜃 𝑡 𝑛 𝑡 cos 𝜃 𝑖 + 𝑛 𝑖 cos 𝜃 𝑡
𝜃 𝑖 = 𝜃 𝐵
Brewster angle (Polarization angle)
𝜃 𝐵 + 𝜃 𝑡 =90°
Γ ⊥ ≠0
Γ || =0
Parallel component only
𝑛 𝑖 sin 𝜃 𝑖 = 𝑛 𝑡 sin 𝜃 𝑡
tan 𝜃 𝐵 = 𝑛 𝑡 / 𝑛 𝑖
𝑛 𝑖 sin 𝜃 𝐵 = 𝑛 𝑡 cos 𝜃 𝐵

10. Total Reflection/Transmission: Water
𝑛 𝑤𝑎𝑡𝑒𝑟 = 𝜀 𝑤𝑎𝑡𝑒𝑟 𝑟𝑒𝑙 ≈ 4 3
tan 𝜃 𝐵 = 𝑛 𝑡 𝑛 𝑖
𝑛 𝑎𝑖𝑟 = 𝜀 𝑎𝑖𝑟 𝑟𝑒𝑙 ≈1
sin 𝜃 𝑐 = 𝑛 𝑡 𝑛 𝑖
Ray travels from
Air to water
Water to air
𝜃 𝑐
N/A
48.59°
𝜃 𝐵
53.13°
36.87°